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Mirrors > Home > MPE Home > Th. List > odmulg2 | Structured version Visualization version GIF version |
Description: The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
odmulgid.1 | ⊢ 𝑋 = (Base‘𝐺) |
odmulgid.2 | ⊢ 𝑂 = (od‘𝐺) |
odmulgid.3 | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
odmulg2 | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘(𝑁 · 𝐴)) ∥ (𝑂‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odmulgid.1 | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
2 | odmulgid.2 | . . . . . 6 ⊢ 𝑂 = (od‘𝐺) | |
3 | 1, 2 | odcl 19528 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ0) |
4 | 3 | nn0zd 12628 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℤ) |
5 | 4 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘𝐴) ∈ ℤ) |
6 | simp3 1135 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
7 | dvdsmul1 16273 | . . 3 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · 𝑁)) | |
8 | 5, 6, 7 | syl2anc 582 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · 𝑁)) |
9 | odmulgid.3 | . . . 4 ⊢ · = (.g‘𝐺) | |
10 | 1, 2, 9 | odmulgid 19546 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℤ) → ((𝑂‘(𝑁 · 𝐴)) ∥ (𝑂‘𝐴) ↔ (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · 𝑁))) |
11 | 5, 10 | mpdan 685 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → ((𝑂‘(𝑁 · 𝐴)) ∥ (𝑂‘𝐴) ↔ (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · 𝑁))) |
12 | 8, 11 | mpbird 256 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘(𝑁 · 𝐴)) ∥ (𝑂‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5144 ‘cfv 6544 (class class class)co 7414 · cmul 11152 ℤcz 12602 ∥ cdvds 16249 Basecbs 17206 Grpcgrp 18921 .gcmg 19055 odcod 19516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-z 12603 df-uz 12867 df-rp 13021 df-fz 13531 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-dvds 16250 df-0g 17449 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-od 19520 |
This theorem is referenced by: odmulgeq 19549 |
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